Journal of Multivariate Analysis 93 (2005) 375–393
Test for parameter change in stochastic processes
based on conditional least-squares estimator
Sangyeol Lee
and Okyoung Na
Department of Statistics, Seoul National University, Seoul, 151-742, South Korea Received 6 December 2002
Abstract
In this paper, we consider the problem of testing for a parameter change in stochastic processes. In performing a test, we employ the cusum test considered in Lee et al. (Scand. J. Statist. 30 (2003) 651). The cusum test is based on the conditional least-squares estimator introduced by Klimko and Nelson (Ann. Statist. 6 (1978) 629). Special attention is paid to the nonlinear autoregressive processes including TAR and ARCH processes. It is shown that under regularity conditions, the test statistic behaves asymptotically the same as the sup of the squares of independent standard Brownian bridges. Simulation results as to ARCH(1) processes and an example of real data analysis are provided for illustration.
r2004 Elsevier Inc. All rights reserved.
AMS 1991 subject classifications: 62M10; 62G10
Keywords: Test for parameter change; Cusum test; Stochastic processes; Nonlinear autoregressive model; Conditional least-squares estimator; Weak convergence; Brownian bridge
1. Introduction
The problem of testing for a parameter change in statistical models has been an important issue among both theoreticians and practitioners. Research into this problem originally began with iid samples; for a review of earlier works, see
[2,4,8,13,17,24]. Subsequently, the issue became very popular in the time series context since time series often suffer from structural changes. Particularly, econometric time series exhibit changes in their underlying model owing to changes
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Corresponding author.
E-mail addresses:[email protected] (S. Lee), [email protected] (O. Na). 0047-259X/$ - see front matter r 2004 Elsevier Inc. All rights reserved.
in governmental policy and critical social events. For references, see
[1,10,12,16,18,21,23]and the papers cited therein.
Incla´n and Tiao[9]and Bai[1]considered the cusum of squares test in testing for a variance and distributional change of random observations. It was proven that it was easy to handle and useful for detecting the locations of change points. The basic idea of the cusum test is somewhat heuristic. For instance, the test for a mean change in an iid sample is based on the following process:
UnðsÞ :¼ 1 ffiffiffi n p s X½ns t¼1 ut ð½ns=nÞ Xn t¼1 ut ! ¼ ½nsffiffiffi n p sð#m½ns#mnÞ; 0psp1; ð1:1Þ
where u1; y; un are iid with mean m and variance s2;and #mn¼ n1Pnt¼1ut:Here, a
large value of sup0psp1jUnðsÞj indicates a change of the mean. The critical values are
obtained asymptotically using the fact that Un converges weakly to a standard
Brownian bridge.
Recently, Lee et al.[15]extended the cusum test to a more general case, motivated by the conjecture: given a parameter of interest and its consistent estimator, under what conditions can the estimator be utilized to detect a change in that parameter. The result of Lee et al.[15]indicates that the cusum test performs well for a broad class of stationary processes including linear processes.
In this article, we focus on the cusum test constructed based on the CLSE (conditional least-squares estimator) (cf.[11]) in stochastic processes. Particularly, special attention is paid to the cusum test in ergodic stationary processes including nonlinear autoregressive (NLAR) processes since they accommodate important nonlinear time series models, such as the autoregressive heteroscedastic (ARCH) and threshold autoregressive (TAR) models, which have been central to the analysis of data with nonlinear characteristics (cf.[22]).
The organization of this paper is as follows. In Section 2, we present the regular conditions under which the cusum test statistic constructed based on the CLSE converges weakly to the sup of the sum of squares for a standard Brownian bridge. In Section 3, as an illustration we consider the parameter change problem in ARCH and TAR processes. Simulation results related to ARCH(1) processes are reported in Section 4. We provide a real data analysis in Section 5.
2. Change point test based on the CLSE 2.1. General case
Let fyt; t¼ 1; 2; yg be a stochastic process defined on a probability space
ðO; F; PyÞ; whose distribution depends on a vector y ¼ ðy1; y;ypÞ0:Herefytg is not
necessarily a stationary process. We denote the true value of y by yo¼ ðyo1; y;yopÞ0:
EjytjoN; tX1; and define the functions gð ; Þ by
gðy; FtÞ ¼ Eyðytþ1jFtÞ; tX0;
where Eyð j Þ is the conditional expectation under Py: Given a set of observations
yt; t¼ 1; y; n; the CLSE #yn¼ ð#y1n; y; #ypnÞ0 of y is defined as the minimizer of the
conditional sum of squares QnðyÞ ¼
Xn1 t¼0
fytþ1 gðy; FtÞg2
and it is actually obtained by solving the least-squares equations
@QnðyÞ=@yi¼ 0; i¼ 1; 2; y; p: ð2:1Þ
Klimko and Nelson[11]showed that there exists a sequence of estimators #ynsuch
that #yn-yo a.s., provided the following regularity conditions are satisfied:
(A1) gðy; FtÞ is twice continuously differentiable with respect to y a.e. in some
neighborhood S of yo:
(A2) For some positive definite (symmetric) matrix V of constants, lim
n-Nð2nÞ 1
Vn¼ V a:s:;
where Vn¼ @2QnðyoÞ=@y2:
(A3) For each 1pi; jpp; lim sup
n-N lim supd-0 ysupANd
ðjTnðyÞijj=ndÞoN a:s:;
where TnðyÞ ¼ @2QnðyÞ=@y2 Vn and Nd is the open sphere of radius d
centered at yo: (A4) For each 1pipp;
lim
n-Nn 1@Q
nðyoÞ=@yi¼ 0 a:s:
Now we consider the problem of testing H0 : y does not change over y1; y; yn vs.
H1 : not H0:
Towards this end, we investigate the asymptotic behavior of the stochastic process VnðsÞ ¼ ½ns ffiffiffi n p ð#y½ns #ynÞ; 0psp1;
from which the test statistic is generated (see (2.13) below). Note that the stochastic process Vn is analogously defined to Un in (1.1).
Assume that Conditions (A1)–(A4) hold. Since f#yng satisfies the least-squares
equations (2.1), we obtain 0¼ n1=2@Qnð#ynÞ=@y
¼ n1=2@QnðyoÞ=@y þ n1ðVnþ TnðynÞÞn1=2ð#yn yoÞ;
where yn¼ yðy1; y; ynÞ is an appropriate intermediate point between yoand #yn;by
expanding the vector n1=2@Qnð#ynÞ=@y in a Taylor series about yo:If there exists an
inverse matrix ofðVnþ TnðynÞÞ; we can write
ffiffiffi n p ð#yn yoÞ ¼ fn1ðVnþ TnðynÞÞg 1 n1=2@QnðyoÞ=@y and consequently, V pffiffiffinð#yn yoÞ ¼ 1 2pffiffiffin @QnðyoÞ @y ðV RnÞR 1 n 1 2pffiffiffin @QnðyoÞ @y ; where Rn¼ ð2nÞ1ðVnþ TnðynÞÞ: Otherwise, we have
V pffiffiffinð#yn yoÞ ¼ 1 2pffiffiffin @QnðyoÞ @y þ ðV RnÞ ffiffiffi n p ð#yn yoÞ: Hence, for 0psp1; V ½nsffiffiffi n p ð#y½ns #ynÞ ¼ 1 2pffiffiffin @Q½nsðyoÞ @y ½ns n 1 2pffiffiffin @QnðyoÞ @y þ ffiffiffiffiffiffiffi ½ns n r D½ns ½ns n Dn; ð2:2Þ where Dk¼ ðV RkÞR1k ð2 ffiffiffi k p
Þ1@QkðyoÞ=@y if R1k exists;
ðV RkÞ ffiffiffi k p ð#yk yoÞ otherwise: (
Now suppose that
1 2pffiffiffin @Q½nsðyoÞ @y ! w W1=2BpðsÞ ð2:3Þ
for some positive definite matrix W ; where Bp¼ ðB1; y; BpÞ0 denotes a
p-dimensional standard Brownian motion. Since
1 2pffiffiffin @Q½nsðyoÞ @y ¼ 1 ffiffiffi n p X ½ns1 t¼0 @gðyo; FtÞ @y fytþ1 gðy o ; FtÞg; f@gðyo; F
tÞ=@y; Ftþ1; tX0g is a predictable sequence and fytþ1
gðyo; FtÞ; Ftþ1; tX0g is a sequence of martingale differences, (2.3) may be
guaranteed by using the functional central limit theorem for martingales (cf.[7]). Note that due to (A2) and (A3), Rn converges to a positive definite matrix V
almost surely as n-N: Therefore, it follows from Egorov’s theorem that given e40 and d40; there exists an event E with PðEÞ41 e=3; a positive real number Z; and a
positive integer N0;such that on E and for n4N0;
lnXZ; ð2:4Þ
where ln denotes the minimum eigenvalue of Rn;and for each 1pi; jpp;
jVij ðRnÞijjpdZ: ð2:5Þ
Since (2.4) implies the existence of an inverse matrix of Rn;we have that on E and for
all n4N0; Dn¼ ðV RnÞR1n ð2 ffiffiffi n p Þ1@QnðyoÞ=@y: Thus, on E; max N0okpn ffiffiffi k p ffiffiffi n p jjDkjj ¼ max N0okpn ffiffiffi k p ffiffiffi n p ðV RkÞR1k 1 2pffiffiffik @QkðyoÞ @y p max N0okpn jjV Rkjj max N0okpn jj R1k jj max N0okpn 1 2pffiffiffin @QkðyoÞ @y pX p i¼1 max N0okpn jjV Rkjj max N0okpn jjR1k jj max 1pkpn 1 2pffiffiffin @QkðyoÞ @yi : ð2:6Þ
First, from (2.5) it holds that on E and for n4N0;
jjV Rnjj :¼ supfjjðV RnÞhjj : jjhjjp1g ¼ sup X p i¼1 Xp j¼1 Vij ðRnÞij hj !2 0 @ 1 A 1=2 : X p j¼1 h2jp1 8 > < > : 9 > = > ; p dZp and consequently, on E max N0okpn jjV RkjjpdZp: ð2:7Þ
Next, since Rnis a real symmetric matrix for all nX1 and (2.4) holds on E and for
all n4N0;we have max N0okpn jjR1 k jj ¼ max N0okpn
fthe maximum absolute eigenvalue of R1 k g
¼ max
N0okpn l1k
Therefore, from (2.7) and (2.8) it follows that the right-hand side of (2.6) is no more thanPpi¼1dp max1pkpnjð2pnffiffiffiÞ1@QkðyoÞ=@yij on E; and consequently,
P max Nokpn ffiffiffi k p ffiffiffi n p jjDkjj4 e 2 ! \ E ( ) p P X p i¼1 dp max 1pkpn 1 2pffiffiffin @QkðyoÞ @yi 4e2 ( ) pX p i¼1 P max 1pkpn @QkðyoÞ @yi 4e ffiffiffi n p dp2 pX p i¼1 d2p4 e2 1 nE @QnðyoÞ @yi 2 ; ð2:9Þ
since f@QnðyoÞ=@yi; Fn; nX1g is a martingale for 1pipp: Here, we can make the
right-hand side of (2.9) no more than e=3 by assuming that Xp
i¼1
n1Eð@QnðyoÞ=@yiÞ2pM for some 0oMoN; ð2:10Þ
and choosing dppffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie3=3p4M:Since there exists a positive integer N4N
0;such that P max 1pkpN0 ffiffiffi k p ffiffiffiffiffi N p jjDkjj4 e 2 ( ) pe 3; we have that for all n4N;
P max 1pkpn ffiffiffi k p ffiffiffi n p jjDkjj4e ( ) p P max 1pkpN0 ffiffiffi k p ffiffiffi n p jjDkjj þ max N0okpn ffiffiffi k p ffiffiffi n p jjDkjj4e ( ) p P max 1pkpN0 ffiffiffi k p ffiffiffi n p jjDkjj4 e 2 ( ) þ P max N0okpn ffiffiffi k p ffiffiffi n p jjDkjj4 e 2 ! \ E ( ) þ PfEcgpe; which implies max 1pkpn ffiffiffi k p ffiffiffi n p jjDkjj ¼ oPð1Þ: ð2:11Þ
Therefore, under the assumptions made in (2.3) and (2.10), we have V ½nsffiffiffi
n
p ð#y½ns #ynÞ ! w
where BopðsÞ ¼ ðBo
1ðsÞ; y; BopðsÞÞ0 is a p-dimensional standard Brownian bridge. The
arguments described above are summarized as follows.
Theorem 2.1. Suppose that Conditions (A1)–(A4), (2.3) and (2.10) hold. Then, W1=2V½nsffiffiffi
n
p ð#y½ns #ynÞ ! w
BopðsÞ:
The above result is immediately applicable to testing H0 vs. H1;we reject H0 if
Tn¼ max 1pkpn k2 nð#yk #ynÞ 0V ˆˆW1Vð#yˆ k #ynÞ ð2:13Þ
is large, where ˆW and ˆV are certain consistent estimators of W and V ; respectively. Since Tnbehaves asymptotically the same as sup0psp1jjBopðsÞjj
2
under H0;the critical
region is given asðTnXCaÞ for nominal level a; where Cais theð1 aÞ-quantile point
of sup0psp1jjBo pðsÞjj
2:
The critical values are simulated by Lee and Park [16]. For instance, when p¼ 2; Ca¼ 3:269; 2.408 and 2.054 for a ¼ 0:01; 0.05 and 0.1,
respectively.
Remark. Although a broad class of time series models satisfy the conditions in Theorem 2.1, for instance, TAR and ARCH models, some important models, such as the polynomial regression model considered in Jandhyala and MacNeill[14], do not satisfy the conditions. Therefore, in advance of applying the result of Theorem 2.1 one must check carefully whether the model under consideration satisfy them.
2.2. Stationary case
Now, we deal with the change point problem in ergodic stationary processes including the NLAR processes. We aim to find some sufficient conditions under which Conditions (A1)–(A4) in Section 2 hold; those with (2.3) and (2.10) will be shown to guarantee an asymptotic result similar to Theorem 2.1.
Let fyt; t¼ 0; 1; yg be an ergodic stationary sequence of integrable r.v.’s, and
assume that there exist a positive integer m such that
Eðytjyt1; y; y0Þ ¼ Eðytjyt1; y; ytmÞ a:s:; tXm: ð2:14Þ
Let Fmt ¼ sðyt; yt1; y; ytmþ1Þ; tXm 1; and let
QnðyÞ ¼
Xn t¼m
fyt gðy; Fmt1Þg 2:
As before, we set gðy; Fm
t1Þ ¼ EyðytjFmt1Þ and denote utðyÞ ¼ yt gðy; Fmt1Þ; tXm:
Suppose that the function g¼ gðy; Fm
t Þ satisfies the following regularity
(R1) For 1pi; j; kpp;
@g=@yi; @2g=@yi@yj and @3g=@yi@yj@yk
exist and are continuous for all yAS; where S is an open neighborhood of yo: (R2) For 1pi; jpp;
Ejum @g=@yijoN; Ejum @2g=@yi@yjjoN; Ej@g=@yi @g=@yjjoN;
where g and its partial derivatives are evaluated at yoand Fmm1:
(R3) For 1pi; j; kpp; there exist functions
Hð0Þðyt; y; ytmþ1Þ; Hið1Þðyt; y; ytmþ1Þ;
Hijð2Þðyt; y; ytmþ1Þ; Hijkð3Þðyt; y; ytmþ1Þ;
such that jgjpHð0Þ; j@g=@y ijpHið1Þ; j@2g=@yi@yjjpH ð2Þ ij ; j@3g=@yi@yj@kjpH ð3Þ ijk
for all yAS; and
Ejym Hijkð3Þðym1; y; y0ÞjoN;
EfHð0Þðy
m1; y; y0Þ Hijkð3Þðym1; y; y0ÞgoN;
EfHið1Þðym1; y; y0Þ Hjkð2Þðym1; y; y0ÞgoN:
Let V be the p p matrix with ði; jÞth component
Eð@gðyo; Fmm1Þ=@yi @gðyo; Fmm1Þ=@yjÞ; ð2:15Þ
which is assumed to be positive definite, and let W be the p p matrix with ði; jÞth component
Eðu2 mðy
oÞ@gðyo
; Fmm1Þ=@yi @gðyo; Fmm1Þ=@yjÞ: ð2:16Þ
For the existence of W ; we assume that for 1pi; jpp;
Eðu2mðyoÞj@gðyo; Fmm1Þ=@yi @gðyo; Fmm1Þ=@yjjÞoN: ð2:17Þ
It is obvious that if Eðu2 mjF
m
m1Þ40 a.s., the positive definiteness of V implies the
same of W : The following is the main result of this subsection.
Theorem 2.2. Suppose that Conditions (R1)–(R3), (2.14) and (2.17) hold. If V and W are positive definite matrices, then we have
W1=2V½nsffiffiffi n
p ð#y½ns #ynÞ ! w
Proof. LetFt ¼ sðyt; y; y0Þ: Since due to (2.14), we have
gðy; FtÞ ¼ gðy; Fmt Þ a:s:; tXm 1;
it can be easily checked that the ergodicity assumption and Conditions (R1)–(R3) imply (A1)–(A4) and (2.10) in Theorem 2.1. Therefore, in view of Theorem 2.1 it suffices to show that (2.3) holds.
Observe that W1=2 1 2pffiffiffin @Q½nsðyoÞ @y ! w BpðsÞ;
where BpðsÞ is a p-dimensional standard Brownian motion, and W1=2¼ ðoijÞpi;j¼1is
a symmetric matrix satisfying W1=2WW1=2¼ I: Indeed, if we put W1=2 1 2pffiffiffin @Q½nsðyoÞ @y ¼ 1 ffiffiffi n p X ½ns t¼m xt;
where xt ¼ ðxi;tÞpi¼1 ¼
Pp
j¼1oij @gðyo; Ft1Þ=@yj utðyoÞ
p
i¼1; fxtg forms a
se-quence of ergodic, stationary martingale differences with uncorrelated components. Then we have 1 ffiffiffi n p X ½ns t¼m xt ! w BpðsÞ
from the facts (cf.[7, p. 311])
ð1Þ For i ¼ 1; y; p and sA½0; 1; X
½ns
t¼m
Eðx2i;t=njFt1Þ ! P
s: ð2Þ For i ¼ 1; y; p and e40; X
n
t¼m
Eðx2
i;t=n Iðjxi;tj= ffiffiffin
p
4eÞjFt1Þ ! P
0: Consequently, (2.3) follows. This completes the proof. &
If ˆW and ˆV are the consistent estimators of W and V ; we obtain from Theorem 2.2 that ˆ W1=2Vˆ ½nsffiffiffi n p ð#y½ns #ynÞ ! w BopðsÞ:
Then the test is performed based on the test statistic Tn ¼ max mpkpn k2 n ð#yk #ynÞ 0V ˆˆW1Vð#yˆ k #ynÞ: 3. Examples
In this section, we apply the results in Section 2 to the nonlinear processes such as ARCH(1) and TAR(p) processes.
3.1. ARCH(1) model
Letfyt: t¼ 0; 1; yg be the stochastic process satisfying the difference equation:
ytþ1¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b0þ b1y2 t q etþ1; ð3:1Þ
where b040; b1X0 and et are iid r.v.’s with zero mean and unit variance. It is well-known that fytg is geometrically ergodic (cf. [5]). Here, we assume that
b41 Q4j¼1 ð2j 1Þo1 to ensure the moment condition Ejy1j8oN (cf.[6, p. 6]).
Since Eðytþ1jyt; y; y0Þ ¼ 0; we can obtain the CLSE #b ¼ ð #b0; #b1Þ0of b¼ ðb0;b1Þ0
usingfy2
tg: Putting F1t ¼ sðy2tÞ; we have
gðb; F1
tÞ ¼ Eðy2tþ1jF1tÞ ¼ b0þ b1y2t
and
utðbÞ ¼ y2t gðb; F1t1Þ ¼ ðb0þ b1y2t1Þ ðe2t 1Þ:
Therefore, (2.14) holds and it is easy to check that g satisfies Conditions (R1)–(R3). Furthermore, in view of (2.15) and (2.16) we have
V ¼ 1 Ey 2 0 Ey20 Ey40 ! and W ¼ Varðe2 1Þ Eðb0þ b1y2 0Þ 2 Ey2 0ðb0þ b1y20Þ 2 Ey20ðb0þ b1y20Þ 2 Ey40ðb0þ b1y20Þ 2 ! :
Here the expected values in the expression of V and W exist by the moment condition, and V is positive definite. Also, W is positive definite since E½u2
1ðbÞjF 1 0 ¼ ðb0þ b1y2 0Þ 2Varðe2
1Þ40 a.s.. Therefore, if the moment condition holds, then the
result of Theorem 2.2 holds.
Now, we consider the following hypotheses: H0: b is constant over y0; y1; y; yn vs.
H1: not H0:
By solving the least-squares equations (2.1) based on y0; y1; y; yk; k¼ 1; 2; y; n;
we obtain the CLSE #bk¼ ð #b0k; #b1kÞ0as follows:
#b1k¼ kPk1t¼0 y2 ty2tþ1 Pk1 t¼0 y2tþ1 Pk1 t¼0 y2t kPk1t¼0 y4 t Pk1 t¼0 y2t 2 and #b0k¼ k1 Xk1 t¼0 y2tþ1 #b1k k1 Xk1 t¼0 y2t:
Meanwhile, from the ergodic theorem we can see that ˆ V¼ 1 Pn1 t¼0 y2t=n Pn1 t¼0 y2t=n Pn1 t¼0 y4t=n !
is a consistent estimator of V : Moreover, ˆ W¼ Pn1 t¼0 ðy2tþ1 #b0n #b1ny2tÞ 2=n Pn1 t¼0 y2t ðy2tþ1 #b0n #b1ny2tÞ 2=n Pn1 t¼0 y2tðy2tþ1 #b0n #b1ny2tÞ 2=n Pn1 t¼0 y4tðy2tþ1 #b0n #b1ny2tÞ 2=n ! ;
is a consistent estimator of W : Therefore, under H0;
ˆ W1=2Vˆ½nsffiffiffi n p ð #b½ns #bnÞ ! w Bo2ðsÞ: Then, the test statistic is given as follows:
TnARCH ¼ max 1pkpn k2 n ð #bk #bnÞ 0V ˆˆW1Vð #bˆ k #bnÞ: 3.2. TAR models
Let us consider the two regime SETAR model yt¼
f10þ f11yt1þ ? þ f1pytpþ s1et if ytdpr;
f20þ f21yt1þ ? þ f2pytpþ s2et if ytd4r;
(
ð3:2Þ where d is a positive integer, NoroN; fij are unknown parameters, si are
positive numbers, and et are iid r.v.’s with zero mean and unit variance, and
independent of the past observations yt1; yt2; y. Let Fi¼ ðfi0;fi1; y;fipÞ0; i¼
1; 2: Throughout, it is assumed that F1aF2 and dpp: In this example, we focus on
the change of F1 and F2 and assume that r and d are known for simplicity, which,
however, can be estimated in actual practice via utilizing the method in Chan[3]. Let xt ¼ ð1; yt1; y; ytpÞ0 and Xt¼ ðx0tIðytdprÞ; x 0 tIðytd4rÞÞ 0:
Then, Eq. (3.2) can be written as
yt¼ X0tFþ fs1IðytdprÞ þ s2Iðytd4rÞget; ð3:3Þ
where F¼ ðF10;F20Þ0:
Here we assume that
(C1) fytg is geometrically ergodic with a unique invariant measure pð Þ:
(C2) et is absolutely continuous with a uniformly continuous and positive pdf, and
Eðe4 tÞoN:
(C3) fytg is stationary with its marginal pdf denoted by pð Þ: Also, Eðy4tÞoN:
(C4) The autoregressive function is discontinuous, that is, there exists Z¼ ð1; zp1; y; z0Þ0 such thatðF1 F2Þ Za0 and zpd ¼ r:
Indeed, if (C2) holds and maxi¼1;2Ppj¼1jfijjo1; (C1) holds and Eðy4tÞoN:
Putting Fpt ¼ sðyt; y; ytpþ1Þ; we have
gðf; Fpt1Þ ¼ X0tF:
It is not difficult to show that Conditions (R1)–(R3) (2.14) and (2.17) are satisfied in a neighborhood of the true parameter. The matrices V and W in the arguments (2.15) and (2.16) are obtained as follows:
V ¼ diagðV1; V2Þ and W ¼ diagðs12V1;s22V2Þ;
where V1¼ Eðxpx0p IðypdprÞÞ and V2¼ Eðxpx0p IðypdorÞÞ: V1 and V2 are
positive definite and so are V and W :
Now, consider the problem of testing for a change of the parameter F: Given the observations yt; t¼ 0; 1; y; n; we intend to test the following
hypotheses:
H0 : F is constant over y0; y1; y; yn vs.
H1 : not H0:
The CLSE is obtained as follows: #Fk¼ ð #F1k0; #F2k0Þ0 by solving the least-squares
equations (2.1) based on y0; y1; y; yn : for ppkpn;
#F1k¼ Xk t¼p xtxt0IðytdprÞ !1 Xk t¼p xtytIðytdprÞ ! ; #F2k¼ Xk t¼p xtxt0Iðytd4rÞ !1 Xk t¼p xtytIðytd4rÞ ! :
Hence under H0;the result of Theorem 2.2 follows, viz.,
½ns ffiffiffi n p #F½ns #Fn !w diagðs1V 1=2 1 ;s2V 1=2 2 ÞB o 2ðpþ1ÞðsÞ: Note that ˆ V1¼ Xn t¼p xtxt0 IðytdprÞ ! ; ˆ V2¼ Xn t¼p xtxt0 Iðytd4rÞ ! ; #s2 1n¼ ðn1Þ1 X ytdpr ðyt xt0#F1Þ2; n1¼ Xn t¼p IðytdprÞ #s2 2n¼ ðn2Þ1 X ytdor ðyt xt0#F2Þ2; n2¼ n n1 p þ 1
are consistent estimators of V1; V2; s21 and s22; respectively. Therefore, the test
statistic is given as follows:
TnTAR¼ max ppkpn X2 i¼1 k2 n #s2 i ð #Fik #FinÞ0Vˆið #Fik #FinÞ:
4. Simulation result in ARCH(1) process
In this section, we evaluate the performance of the test TARCH
n introduced in
Section 3 through a simulation study. Special attention will be paid to the comparison study of our test and the cusum of squares test considered in[9,10]. This comparison study is of great interest in the aspect that our method tests a change of parameters themselves while the cusum test essentially tests the change of a functional of the parameters. The empirical sizes and powers are calculated at nominal level a¼ 0:05: Here, the critical value at a ¼ 0:05 is 2.408. In this simulation 100 initial observations are discarded to remove initialization effects.
In order to evaluate the performance of the test TARCH
n ;we consider the ARCH(1)
model in (3.1), where etare iid standard normal r.v’s and y0¼ 0: The empirical sizes
are calculated with sets of 500, 800, and 1000 observations generated from the ARCH(1) model with b¼ ðb0;b1Þ ¼ ð0:5; 0:0Þ; (0.5, 0.15), (0.5, 0.3), (1.0, 0.0), (1.0,
0.15) and (1.0, 0.3). The figures inTable 1indicate the proportion of the number of rejections of the null hypothesis H0under which no parameter changes are assumed
to occur, out of 1000 repetitions. The results show that our test has small size distortions unless the parameters are near the boundary of the region on which Ey8
1oN is satisfied, viz., the ðb1¼ 0:3Þ case. From the fact that the size distortion
disappears when n¼ 1000; we can see that the test is stable for data with large sample size.
In order to examine the power we consider the alternative hypothesis H1: b¼ ðb0;b1Þ changes from b0¼ ðb00;b 0 1Þ to b 1¼ ðb1 0;b 1 1Þ at t ¼ ½n=2;
where n denotes the sample size and½n=2 is the point where the parameter change occurs. For n¼ 500; 800, 1000, we consider the two cases: (i) b0remains as 0.5 and
Table 1
Empirical sizes of TARCH
n and Bnð ˆCÞ n b0¼ 0:5 b0¼ 1:0 b1¼ 0:00 b1¼ 0:15 b1¼ 0:30 b1¼ 0:00 b1¼ 0:15 b1¼ 0:30 500 0.078 (0.042) 0.075 (0.038) 0.092 (0.029) 0.080 (0.034) 0.069 (0.039) 0.088 (0.023) 800 0.065 (0.043) 0.067 (0.032) 0.087 (0.028) 0.053 (0.047) 0.073 (0.034) 0.080 (0.030) 1000 0.058 (0.048) 0.043 (0.038) 0.070 (0.037) 0.050 (0.040) 0.057 (0.036) 0.074 (0.034)
b1changes from b01 to b11with b01¼ 0; 0.15, 0.3 and b1
1¼ 0:3; 0.6, 0.9; (ii) b0changes
from 1.0 to 0.5, and b1changes in the same way as in (i). The results are summarized in Tables 2 and 3, and the figures in those tables are calculated as the rejection number of the null hypothesis out of 1000 repetitions. Therein, it can be observed that the test produces good powers and the power of TnARCH increases as either the difference between b0 and b1 or n increases as might be anticipated. Our results enable us to conclude that the test performs well for data with fairly large sample size.
Table 2
Empirical powers of TARCH
n and Bnð ˆCÞ when b1changes from b01to b 1
1and b0¼ 0:5 remains the same
b1 1 b0 1 n 0.3 0.6 0.9 0.00 500 0.697 (0.354) 0.982 (0.603) 1.00 (0.481) 800 0.877 (0.599) 1.00 (0.778) 1.00 (0.610) 1000 0.921 (0.705) 1.00 (0.820) 1.00 (0.674) 0.15 500 0.262 (0.095) 0.841 (0.407) 0.992 (0.448) 800 0.297 (0.163) 0.955 (0.628) 1.00 (0.570) 1000 0.372 (0.203) 0.983 (0.747) 1.00 (0.601) 0.30 500 0.088 (0.023) 0.589 (0.168) 0.960 (0.319) 800 0.080 (0.030) 0.701 (0.286) 0.993 (0.458) 1000 0.074 (0.034) 0.779 (0.393) 0.996 (0.537) Table 3
Empirical powers of TARCH
n and Bnð ˆCÞ when b0¼ ðb0;b1Þ changes from ð1:0; b01Þ to ð0:5; b 1 1Þ b11 b01 n 0.3 0.6 0.9 0.00 500 0.880 (0.509) 0.723 (0.050) 0.905 (0.130) 800 0.981 (0.700) 0.815 (0.045) 0.966 (0.286) 1000 0.991 (0.769) 0.855 (0.065) 0.994 (0.363) 0.15 500 0.878 (0.719) 0.569 (0.084) 0.782 (0.082) 800 0.974 (0.879) 0.615 (0.099) 0.874 (0.159) 1000 0.992 (0.940) 0.643 (0.099) 0.919 (0.204) 0.3 500 0.922 (0.815) 0.498 (0.141) 0.626 (0.044) 800 0.987 (0.960) 0.589 (0.179) 0.687 (0.067) 1000 0.995 (0.983) 0.606 (0.215) 0.716 (0.087)
Next, we perform a comparison study of our test and the cusum of squares test Bnð ˆCÞ ¼ ˆC ffiffiffin p max 1plpn jDlj; where Dl¼ Pl t¼1y2t Pn t¼1y2t l n; ˆ C2¼ð1 #b1Þð1 #k#b 2 1Þ ð#k 1Þð1 þ #b1Þ ; #k ¼ Eyˆ 4t #b2 0þ #b21Eyˆ 4t þ 2 #b20#b1ð1 #b1Þ 1; ˆ Ey4 t ¼ n1 Pn
t¼1y4t and #b0; #b1 are CLSEs. The empirical sizes and powers are
computed for the same situation as above. The figures in the parentheses inTables 1– 3 denote the sizes and powers. As seen in the tables, the cusum of squares test produces less size distortions than our test. However, it produced very poor powers in many cases. It might be unfair to say that our test outperforms the cusum of squares test perfectly taking into consideration the size distortion occurring when no1000; but the gap was still enormous even at n ¼ 1000: In fact, the cusum of squares test in ARCH models has a drawback since it intrinsically detects a change of unconditional variance o :¼ b0=ð1 b1Þ: Note that if b0 decreases and b1
increases at the same time, o may not change substantially. Therefore, it only represents an indirect approach to the detection of changes in the ARCH parameters. The result of this comparison study strongly supports the validity of our approach for the parameter change test. In conclusion, we recommend to utilize our method provided that the regularity conditions are met in given situation.
5. Real data analysis
In this section we analyze a real data set for illustration. Here, we investigate a change point for the quarterly data of the GDP of community, social and personal services in Korea for the period 1971:1–2003:3, namely, 131 observations. For this task, provided that there are no changes, we assume that the transformed data yt ¼ log xtþ1 log xt; t¼ 1; 2; y; 130 (seeFig. 1), where xtdenote the original data,
satisfies the model yt¼ X4 i¼1 miIAiðtÞ þ f yt4 X4 i¼1 miIAiðt 4Þ ! þ et; ð5:1Þ
wherejfjo1; IAi is the indicator of Ai¼ fi; i þ 4; i þ 8; yg; i ¼ 1; y; 4; and et are iid r.v.’s with zero mean and finite variance s2:Obviously, the above model reflects
that the means of the four quarter data sets are non-identical, and so is nonstationary.
Now we show that the model in (5.1) satisfies the regularity conditions in Section 2. Let y¼ ðm1; y;m4;fÞ0and F
t ¼ sðy1; y2; y; ytÞ: Then we have
gðy; Ft1Þ ¼ X4 i¼1 miIAiðtÞ þ f yt4 X4 i¼1 miIAiðt 4Þ !
and gðy; Ft1Þ is three times differentiable with respect to y a.e..
SincefZt :¼ yt P4i¼1miIAiðtÞg satisfies the following difference equation: Zt¼ fZt4þ et; jfjo1;
it is a stationary linear process of the form Zt¼X
N
k¼0
fket4k:
Therefore, by using the strong law of large numbers for linear processes, we can readily check that Qn satisfies Conditions (A2)–(A4), and
V ¼ diagð41ð1 fÞ2; y;41ð1 fÞ2;s2ð1 f2Þ1Þ:
Meanwhile, using the functional central limit theorem for martingales and the Crame´r–Wold device, one can see that
1 2pffiffiffin @Q½nsðy0Þ @y ¼ 1ffiffiffi n p X ½ns t¼1 @gðy0; Ft1Þ @y et ¼ 1ffiffiffi n p X ½ns t¼1 etð1 fÞIA1ðtÞ; y; 1 ffiffiffi n p X ½ns t¼1 etð1 fÞIA1ðtÞ; 1 ffiffiffi n p X ½ns t¼1 etZt4 ! !w s2V B5ðsÞ 0 20 40 60 80 100 120 -4 -2 0 1Q 2Q 3Q 4Q 8 6 4 2
and that X5
i¼1
n1Eð@Qnðy0Þ=@yiÞ2¼ 4s2ð1 fÞ2þ 4s4=ð1 f2ÞoN:
Now that the conditions in Theorem 2.1 hold, we can test for the constancy of y using the test statistic
Tn:¼ max 11pkpnTn;k¼ max11pkpn k2 n ðyk ynÞ 0V ˆˆW1Vðyˆ k ynÞ ¼ max 11pkpn k2 n Xm i¼1 ð#mk;i#mn;iÞ2 ð1 #fnÞ2 m#s2 þ ð #fk #fnÞ 2 1 1 #f2 ( ) ; ð5:2Þ
where #yk¼ ð#mk;1; y;#mk;4; #fkÞ0;11pkpn; denotes the CLSE
#fk¼
Pk4
t¼1 ytþ4ðytP4i¼1mk;iIAiðtÞÞ Pk4
t¼1 ðytP4i¼1mk;iIAiðtÞÞ
2 ; mk;i ¼ Pk4 t¼1 ytIAiðtÞ Pk4 t¼1 IAiðtÞ ; #mk;i¼ Pk4 t¼1 ytþ4IAiðtÞ #fk Pk4 t¼1 ytIAiðtÞ ð1 #fkÞPk4t¼1 IAiðtÞ
and #s2 is the sample variance of the residuals. For constructing T
n; we only
employed Tnk’s for kX11 since initial values of Tn;k’s may seriously mislead the test
according to our previous experience.
Theorem 2.1 indicates that the statistic Tnin (5.2) is approximately distributed as
sup0psp1jjBo 5ðsÞjj
2
under the null hypothesis where no changes are assumed to occur. At the nominal level 0.05, the critical value is 3.899 (cf.[15]: p¼ 5). Since T130¼
75:345 is larger than this number, the null hypothesis is rejected at the 0.05 level. Since the maximum value of T130;kis obtained at k¼ 35 (seeFig. 2), the estimated
change point is k¼ 35; which corresponds to the third quarter of 1979: the vertical line inFig. 1indicates the location of the change point.Table 4presents the estimates
20 40 60 80 100 120 0 20 40 60 k=35, Tnk=75.345 3.899 Fig. 2. Plot of Tn;k:
of y for the two subseries before and after the change point, respectively. From
Table 4, one can see that there are significant differences between the two sets of estimates. Using the Ljung–Box test, available in SAS program, version 8.1, we could check that both subseries follow the model in (5.1). This result demonstrates the validity of our test.
Acknowledgments
We thank the two referees for valuable comments that improve the quality of the paper. We wish to acknowledge that this research was supported (in part) by KOSEF through the Statistical Research Center for Complex Systems at Seoul National University.
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